# Finite morphism “with the right number of equations”

Suppose that $X$ is a scheme, say smooth, connected of finite type over a field of characteristic zero, $Y \to X$ is a smooth morphism of relative dimension $d$, and $f_1, \dots, f_d \in \mathcal{O}(Y)$ are $d$ global sections. Let $Z = Z(f_1, \dots, f_d) \subset Y$ be the closed (not necessarily reduced) subscheme defined by the global sections. Assume that the composite $Z \to X$ is finite.

What can be concluded in this situation? As far as I understand if $Z$ happened to be integral then it would be Cohen-Macaylay and flat over $X$. Is there something that can be said in greater generality? What if I assume that $X$ is the spectrum of a Henselian local ring, for example?

Related question: Are finite correspondances flat?

• Your assumption that $Z \to X$ is finite implies that $Z$ is lci (since $X$ and $Y$ are smooth). So, $Z$ is also Cohen--Macaulay, hence it is flat over $X$ without any further assumptions. – ulrich Sep 6 '16 at 10:22
• Are you saying that the $f_i$ automatically form a regular sequence? Do you have a reference? I am afraid my ignorance of geometry is showing :/. – Tom Bachmann Sep 6 '16 at 10:47
• Yes, the $f_i$ indeed form a regular sequence. See, for example, Theorem 2.12 (c) of the book "Cohen-Macaulay rings" by Bruns--Herzog (or any other commutative algebra book which discusses regular sequences). – ulrich Sep 7 '16 at 5:42
• Fantastic, thank you! If you post this as an answer I will accept it. – Tom Bachmann Sep 7 '16 at 9:17
• (For anyone else looking, it's theorem 2.1.2 (c) in the book cited.) – Tom Bachmann Sep 7 '16 at 9:53